W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published:17 May 2013

views:61507

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

published:01 May 2013

views:24132

published:13 May 2016

views:773

The All-New Land Rover Discovery receives the surfing seal of approval as legendary surfer Laird Hamilton and 8-year old surfing prodigy Jett Prefontaine take the ultimate family SUV on a surf adventure in Malibu. In a display of incredible versatility, smart technology and capability, the All-New Discovery proves why it’s the perfect SUV for any adventure.
Visithttp://bit.ly/2fBmbtf for more All-New Discovery Information.

published:15 Nov 2016

views:28053

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier.
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
http://www.history.com
Check out exclusive HISTORY content:
Website - http://www.history.com
Facebook - https://www.facebook.com/History
Twitter - https://twitter.com/history
Google+ - https://plus.google.com/+HISTORY
HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect viewers with history in an informative, immersive, and entertaining manner across all platforms. The network’s all-original programming slate features a roster of hit series, epic miniseries, and scripted event programming. Visit us at HISTORY.com for more info.

published:07 Jul 2017

views:74375

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published:09 Aug 2013

views:13306

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

published:24 Jul 2013

views:13855

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

published:01 Dec 2015

views:5636

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

published:17 May 2016

views:10

For more information please go to https://www.lodge.co.nz/Browse-Properties/Flagstaff/20-Discovery-Drive-Flagstaff/LF06951

In September 2013 Jaguar Land Rover announced plans to open a 100 million GBP (160 million USD) research and development centre in the University of Warwick, Coventry to create a next generation of vehicle technologies. The carmaker said around 1,000 academics and engineers would work there and that construction would start in 2014.

Land Rover Discovery

The Land Rover Discovery is a mid-size luxury SUV, from the British car maker Land Rover. There have been two generations of the vehicle, the first of which was introduced in 1989 and given a Series II update in 1998. The second generation, titled Discovery 3, launched in 2004 and was marketed in North America as the Land Rover LR3. These second generation models were updated in 2009 as the Discovery 4—Land Rover LR4 for North American markets.

First generation

Discovery Series I (1989–1998)

The Discovery Series I was introduced into the United Kingdom in 1989. The company code-named the vehicle "Project Jay". The new model was based on the chassis and drivetrain of the more upmarket Range Rover, but with a lower price aimed at a larger market segment intended to compete with Japanese offerings. This was the only Discovery generation with a four-cylinderpetrol engine.

The Discovery was initially only available as a three-door version; the five-door body style became available in 1990. Both were fitted with five seats, with the option to have two jump seats fitted in the boot. Land Rover employed an external consultancy, Conran Design Group, to design the interior. They were instructed to ignore current car interior design and position the vehicle as a 'lifestyle accessory'. Their interior incorporated a number of original features, although some ideas shown on the original interior mock-ups (constructed inside a Range Rover bodyshell at Conran's workshops) were left on the shelf, such as a custom sunglasses holder built into the centre of the steering wheel. The design was unveiled to critical acclaim, and won a British Design Award in 1989.

FamousMathProbs 13a: The rotation problem and Hamilton's discovery of quaternions I

FamousMathProbs 13a: The rotation problem and Hamilton's discovery of quaternions I

FamousMathProbs 13a: The rotation problem and Hamilton's discovery of quaternions I

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

59:47

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

5:55

Possible cave discovery ,May 2016 .Hamilton Ontario Canada .

Possible cave discovery ,May 2016 .Hamilton Ontario Canada .

Possible cave discovery ,May 2016 .Hamilton Ontario Canada .

1:45

All-New Land Rover Discovery – Surfing with Laird Hamilton

All-New Land Rover Discovery – Surfing with Laird Hamilton

All-New Land Rover Discovery – Surfing with Laird Hamilton

The All-New Land Rover Discovery receives the surfing seal of approval as legendary surfer Laird Hamilton and 8-year old surfing prodigy Jett Prefontaine take the ultimate family SUV on a surf adventure in Malibu. In a display of incredible versatility, smart technology and capability, the All-New Discovery proves why it’s the perfect SUV for any adventure.
Visithttp://bit.ly/2fBmbtf for more All-New Discovery Information.

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier.
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
http://www.history.com
Check out exclusive HISTORY content:
Website - http://www.history.com
Facebook - https://www.facebook.com/History
Twitter - https://twitter.com/history
Google+ - https://plus.google.com/+HISTORY
HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect viewers with history in an informative, immersive, and entertaining manner across all platforms. The network’s all-original programming slate features a roster of hit series, epic miniseries, and scripted event programming. Visit us at HISTORY.com for more info.

1:01:33

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

56:14

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

54:56

Hamilton, Boole and their Algebras - Professor Raymond Flood

Hamilton, Boole and their Algebras - Professor Raymond Flood

Hamilton, Boole and their Algebras - Professor Raymond Flood

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

4:47

Forbes Hamilton, Land Rover Discovery

Forbes Hamilton, Land Rover Discovery

Forbes Hamilton, Land Rover Discovery

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

1:46

20 Discovery Drive, Flagstaff, Hamilton.

20 Discovery Drive, Flagstaff, Hamilton.

20 Discovery Drive, Flagstaff, Hamilton.

For more information please go to https://www.lodge.co.nz/Browse-Properties/Flagstaff/20-Discovery-Drive-Flagstaff/LF06951

Hamilton Discovery School

2003 Land Rover Discovery SE for sale in Hamilton, OH

Click here to learn more about this vehicle: http://www.internationalautooutlet.com/vehicle-details/51e9e82a1774b740873bc3ad536e9fae/default.html2003Land Rover Discovery SE--IMMACULATE--4WD--LEATHER INTERIOR--HEATED SEATS--POWER OPTIONS--A MUST SEE TO APPRECIATE--MOST OF OUR VEHICLES ARE HIGH QUALITY, HAND PICKED, ONE OWNER IN A LIKE NEW CONDITION WITH A CLEANCARFAX. ALL ARE FULLY INSPECTED, SERVICED AND RECONDITIONED, THOSE THAT DO NOT MEET OUR MECHANICAL CRITERIA ARE NOT OFFERED FOR SALE. MOST OF OUR VEHICLES ARE COVERED WITH THE MANUFACTURER WARRANTY OR A 3 MONTHS/4500 MILE WARRANTY. FINANCING IS AVAILABLE AND TRADES ARE ALWAYS WELCOMED. FOR SIMILAR GREAT DEALS PLEASE VISIT OUR WEBSITE http://www.InternationalAutoOutlet.com
Click here to visit our website: http://www.internationalautooutlet.com

2016 Discovery Days in Health Sciences Hamilton

Our 4th annual Discovery Day in Health Sciences hosted by Hamilton Health Sciences gave area high school students and teachers another opportunity to explore careers in medicine and other sciences via a keynote lecture, hands-on workshops and an interactive lab demo which concluded the day. Take a peak at some of the things we did and share the excitement!

Deep Sea Turtle Discovery Hamilton Island 2016

FamousMathProbs 13a: The rotation problem and Hamilton's discovery of quaternions I

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at ...

published: 17 May 2013

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors ...

published: 01 May 2013

Possible cave discovery ,May 2016 .Hamilton Ontario Canada .

published: 13 May 2016

All-New Land Rover Discovery – Surfing with Laird Hamilton

The All-New Land Rover Discovery receives the surfing seal of approval as legendary surfer Laird Hamilton and 8-year old surfing prodigy Jett Prefontaine take the ultimate family SUV on a surf adventure in Malibu. In a display of incredible versatility, smart technology and capability, the All-New Discovery proves why it’s the perfect SUV for any adventure.
Visithttp://bit.ly/2fBmbtf for more All-New Discovery Information.

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier.
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
http://www.history.com
Check out exclusive HISTORY content:
Website - http://www.history.com
Facebook - https://www.facebook.com/History
Twitter - https://twitter.com/history
Google+ - https://plus.google.com/+HISTORY
HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect view...

published: 07 Jul 2017

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vecto...

published: 09 Aug 2013

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slow...

published: 24 Jul 2013

Hamilton, Boole and their Algebras - Professor Raymond Flood

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-...

published: 01 Dec 2015

Forbes Hamilton, Land Rover Discovery

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

published: 17 May 2016

20 Discovery Drive, Flagstaff, Hamilton.

For more information please go to https://www.lodge.co.nz/Browse-Properties/Flagstaff/20-Discovery-Drive-Flagstaff/LF06951

Hamilton Discovery School

Hamilton Discovery School

2003 Land Rover Discovery SE for sale in Hamilton, OH

Click here to learn more about this vehicle: http://www.internationalautooutlet.com/vehicle-details/51e9e82a1774b740873bc3ad536e9fae/default.html2003Land Rover Discovery SE--IMMACULATE--4WD--LEATHER INTERIOR--HEATED SEATS--POWER OPTIONS--A MUST SEE TO APPRECIATE--MOST OF OUR VEHICLES ARE HIGH QUALITY, HAND PICKED, ONE OWNER IN A LIKE NEW CONDITION WITH A CLEANCARFAX. ALL ARE FULLY INSPECTED, SERVICED AND RECONDITIONED, THOSE THAT DO NOT MEET OUR MECHANICAL CRITERIA ARE NOT OFFERED FOR SALE. MOST OF OUR VEHICLES ARE COVERED WITH THE MANUFACTURER WARRANTY OR A 3 MONTHS/4500 MILE WARRANTY. FINANCING IS AVAILABLE AND TRADES ARE ALWAYS WELCOMED. FOR SIMILAR GREAT DEALS PLEASE VISIT OUR WEBSITE http://www.InternationalAutoOutlet.com
Click here to visit our website: http://www.internationalau...

2016 Discovery Days in Health Sciences Hamilton

Our 4th annual Discovery Day in Health Sciences hosted by Hamilton Health Sciences gave area high school students and teachers another opportunity to explore careers in medicine and other sciences via a keynote lecture, hands-on workshops and an interactive lab demo which concluded the day. Take a peak at some of the things we did and share the excitement!

Deep Sea Turtle Discovery Hamilton Island 2016

FamousMathProbs 13a: The rotation problem and Hamilton's discovery of quaternions I

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with h...

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published:17 May 2013

views:61507

back

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem ...

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

The All-New Land Rover Discovery receives the surfing seal of approval as legendary surfer Laird Hamilton and 8-year old surfing prodigy Jett Prefontaine take the ultimate family SUV on a surf adventure in Malibu. In a display of incredible versatility, smart technology and capability, the All-New Discovery proves why it’s the perfect SUV for any adventure.
Visithttp://bit.ly/2fBmbtf for more All-New Discovery Information.

The All-New Land Rover Discovery receives the surfing seal of approval as legendary surfer Laird Hamilton and 8-year old surfing prodigy Jett Prefontaine take the ultimate family SUV on a surf adventure in Malibu. In a display of incredible versatility, smart technology and capability, the All-New Discovery proves why it’s the perfect SUV for any adventure.
Visithttp://bit.ly/2fBmbtf for more All-New Discovery Information.

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier.
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
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Check out exclusive HISTORY content:
Website - http://www.history.com
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HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect viewers with history in an informative, immersive, and entertaining manner across all platforms. The network’s all-original programming slate features a roster of hit series, epic miniseries, and scripted event programming. Visit us at HISTORY.com for more info.

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier.
Subscribe for more from HISTORY:
http://www.youtube.com/subscription_center?add_user=historychannel
Find out more about this and other specials on our site:
http://www.history.com
Check out exclusive HISTORY content:
Website - http://www.history.com
Facebook - https://www.facebook.com/History
Twitter - https://twitter.com/history
Google+ - https://plus.google.com/+HISTORY
HISTORY SpecialsSeason 1Episode 1
THE HISTORY CHANNEL brings history's most incredible wartime feats, scientific mysteries, and turbulent periods back to life.
HISTORY®, now reaching more than 98 million homes, is the leading destination for award-winning original series and specials that connect viewers with history in an informative, immersive, and entertaining manner across all platforms. The network’s all-original programming slate features a roster of hit series, epic miniseries, and scripted event programming. Visit us at HISTORY.com for more info.

published:07 Jul 2017

views:74375

back

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion o...

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published:09 Aug 2013

views:13306

back

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discove...

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

Click here to learn more about this vehicle: http://www.internationalautooutlet.com/vehicle-details/51e9e82a1774b740873bc3ad536e9fae/default.html2003Land Rover Discovery SE--IMMACULATE--4WD--LEATHER INTERIOR--HEATED SEATS--POWER OPTIONS--A MUST SEE TO APPRECIATE--MOST OF OUR VEHICLES ARE HIGH QUALITY, HAND PICKED, ONE OWNER IN A LIKE NEW CONDITION WITH A CLEANCARFAX. ALL ARE FULLY INSPECTED, SERVICED AND RECONDITIONED, THOSE THAT DO NOT MEET OUR MECHANICAL CRITERIA ARE NOT OFFERED FOR SALE. MOST OF OUR VEHICLES ARE COVERED WITH THE MANUFACTURER WARRANTY OR A 3 MONTHS/4500 MILE WARRANTY. FINANCING IS AVAILABLE AND TRADES ARE ALWAYS WELCOMED. FOR SIMILAR GREAT DEALS PLEASE VISIT OUR WEBSITE http://www.InternationalAutoOutlet.com
Click here to visit our website: http://www.internationalautooutlet.com

Click here to learn more about this vehicle: http://www.internationalautooutlet.com/vehicle-details/51e9e82a1774b740873bc3ad536e9fae/default.html2003Land Rover Discovery SE--IMMACULATE--4WD--LEATHER INTERIOR--HEATED SEATS--POWER OPTIONS--A MUST SEE TO APPRECIATE--MOST OF OUR VEHICLES ARE HIGH QUALITY, HAND PICKED, ONE OWNER IN A LIKE NEW CONDITION WITH A CLEANCARFAX. ALL ARE FULLY INSPECTED, SERVICED AND RECONDITIONED, THOSE THAT DO NOT MEET OUR MECHANICAL CRITERIA ARE NOT OFFERED FOR SALE. MOST OF OUR VEHICLES ARE COVERED WITH THE MANUFACTURER WARRANTY OR A 3 MONTHS/4500 MILE WARRANTY. FINANCING IS AVAILABLE AND TRADES ARE ALWAYS WELCOMED. FOR SIMILAR GREAT DEALS PLEASE VISIT OUR WEBSITE http://www.InternationalAutoOutlet.com
Click here to visit our website: http://www.internationalautooutlet.com

Our 4th annual Discovery Day in Health Sciences hosted by Hamilton Health Sciences gave area high school students and teachers another opportunity to explore careers in medicine and other sciences via a keynote lecture, hands-on workshops and an interactive lab demo which concluded the day. Take a peak at some of the things we did and share the excitement!

Our 4th annual Discovery Day in Health Sciences hosted by Hamilton Health Sciences gave area high school students and teachers another opportunity to explore careers in medicine and other sciences via a keynote lecture, hands-on workshops and an interactive lab demo which concluded the day. Take a peak at some of the things we did and share the excitement!

FamousMathProbs 13a: The rotation problem and Hamilton's discovery of quaternions I

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at ...

published: 17 May 2013

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors ...

published: 01 May 2013

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vecto...

published: 09 Aug 2013

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slow...

published: 24 Jul 2013

Hamilton, Boole and their Algebras - Professor Raymond Flood

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-...

ALEXANDER HAMILTON

Fentanyl: The Drug Deadlier than Heroin

VICE presents an immersive and personal feature film about the fentanyl crisis in Canada told from the perspective of a community of drug users.
This video has been edited from its original version due to privacy concerns surrounding one of its subjects.
WATCH NEXT:
Black Widow: Dominating the MasculineWorld of Boxing: http://bit.ly/28UgNhS
Watch more of our documentaries on drugs: http://bit.ly/2aB0DuG
Click here to subscribe to VICE: http://bit.ly/Subscribe-to-VICE
Check out our full video catalog: http://bit.ly/VICE-Videos
Videos, daily editorial and more: http://vice.com
More videos from the VICE network: https://www.fb.com/vicevideo
Like VICE on Facebook: http://fb.com/vice
Follow VICE on Twitter: http://twitter.com/vice
Read our Tumblr: http://vicemag.tumblr.com
Follow us on In...

published: 22 Jul 2016

America Before Columbus (Full Documentary)

History books traditionally depict the pre-Columbus Americas as a pristine wilderness where small native villages lived in harmony with nature. But scientific evidence tells a very different story: When Columbus stepped ashore in 1492, millions of people were already living there. America wasn't exactly a New World, but a very old one whose inhabitants had built a vast infrastructure of cities, orchards, canals and causeways.
The English brought honeybees to the Americas for honey, but the bees pollinated orchards along the East Coast. Thanks to the feral honeybees, many of the plants the Europeans brought, like apples and peaches, proliferated. Some 12,000 years ago, North American mammoths, ancient horses, and other large mammals vanished. The first horses in America since the Pleistoce...

Formula 1: The Limit HD [Documentary]

How I install a pocket watch mainspring, Hamilton 940, going barrel

Watch don't work because mainspring is broken.
This video performed by an amateur. Do not attempt at home. No watches were harmed in the making of this video. For entertainment only.

published: 16 Oct 2017

Tyler Hamilton - The true price of winning at all costs

The Discovery Vitality Summit, held from 15-17 August this year, featured a number of keynote speakers, both local and international. One of the main draw cards was Tyler Hamilton, former professional cyclist and teammate of Lance Armstrong. In his talk, Tyler opened up about his journey and the choices he made that led him to become a cycling champion and a user of doping substances. He also fielded questions from the audience about his experience. The footage is an edited version of his talk.

Formula One - Secret Life HD [Full Documentary]

Lick Observatory: Over 100 Years of Discovery

Lick Observatory, the first permanently occupied mountaintop observatory, has been at the forefront of astronomical research for more than 100 years. Enjoy an insider's tour of Mt. Hamilton's major telescopes and hear Lick's astronomers and astrophysicists talk about the Observatory's remarkable accomplishments - from early discoveries to modern day research. [8/2001] [Science] [Show ID: 5928]

published: 31 Jan 2008

The Hamilton-LaRouche Secret to Economics

While Hillary and Trump distract from serious policy issues that need to be debated in this presidential election, Lyndon LaRouche is the only member of the U.S. Presidency organizing for a Hamiltonian recovery program. See his "The Four New Laws to SaveThe U.S.A. Now! NOT AN OPTION: AN IMMEDIATE NECESSITY"(https://larouchepac.com/four-laws).
Here is an excerpt of our October 14, 2016 webcast featuring MatthewOgden and Ben Deniston discussing the unique genius of Alexander Hamilton and how his approach to growing the economy can be applied to the United States today. Full webcast: https://www.youtube.com/watch?v=jm1nqnvPCEs
----------
Subscribe to LaRouchePAC Live: http://lpac.co/youtube
Subscribe to LaRouchePAC Videos: http://lpac.co/youtube-vid
Subscribe to LaRouchePAC Science: ...

published: 17 Oct 2016

Liberty's Kids HD 120 - Alexander Hamilton - An American In Paris | History Cartoons for Children

“There’s a million things I haven’t done, but just you wait, JUST YOU WAIT!"
Subscribe to Liberty's Kids : http://bit.ly/2ivIdi6
Benjamin Franklin enlists the help of young people to record the happenings leading up to and during the Revolution for his newspaper the Pennsylvania Gazette. First, there is a James Hiller, a patriot who tends to act before he thinks and is, at times, too quick a judge in his search for American heroes. Next is Henri, a French orphan who's only quest is for food. Lastly, a young woman named Sarah Phillips joins the team; the daughter of an ex-English general with strong views opposing slavery. Together, they travel the colonies witnessing the sacrifices made for freedom.

FamousMathProbs 13a: The rotation problem and Hamilton's discovery of quaternions I

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with h...

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published:17 May 2013

views:61507

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FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem ...

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

published:01 May 2013

views:24132

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FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion o...

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

published:09 Aug 2013

views:13306

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FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discove...

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

Fentanyl: The Drug Deadlier than Heroin

VICE presents an immersive and personal feature film about the fentanyl crisis in Canada told from the perspective of a community of drug users.
This video has...

VICE presents an immersive and personal feature film about the fentanyl crisis in Canada told from the perspective of a community of drug users.
This video has been edited from its original version due to privacy concerns surrounding one of its subjects.
WATCH NEXT:
Black Widow: Dominating the MasculineWorld of Boxing: http://bit.ly/28UgNhS
Watch more of our documentaries on drugs: http://bit.ly/2aB0DuG
Click here to subscribe to VICE: http://bit.ly/Subscribe-to-VICE
Check out our full video catalog: http://bit.ly/VICE-Videos
Videos, daily editorial and more: http://vice.com
More videos from the VICE network: https://www.fb.com/vicevideo
Like VICE on Facebook: http://fb.com/vice
Follow VICE on Twitter: http://twitter.com/vice
Read our Tumblr: http://vicemag.tumblr.com
Follow us on Instagram: http://instagram.com/vice
Check out our Pinterest: https://pinterest.com/vicemag

VICE presents an immersive and personal feature film about the fentanyl crisis in Canada told from the perspective of a community of drug users.
This video has been edited from its original version due to privacy concerns surrounding one of its subjects.
WATCH NEXT:
Black Widow: Dominating the MasculineWorld of Boxing: http://bit.ly/28UgNhS
Watch more of our documentaries on drugs: http://bit.ly/2aB0DuG
Click here to subscribe to VICE: http://bit.ly/Subscribe-to-VICE
Check out our full video catalog: http://bit.ly/VICE-Videos
Videos, daily editorial and more: http://vice.com
More videos from the VICE network: https://www.fb.com/vicevideo
Like VICE on Facebook: http://fb.com/vice
Follow VICE on Twitter: http://twitter.com/vice
Read our Tumblr: http://vicemag.tumblr.com
Follow us on Instagram: http://instagram.com/vice
Check out our Pinterest: https://pinterest.com/vicemag

America Before Columbus (Full Documentary)

History books traditionally depict the pre-Columbus Americas as a pristine wilderness where small native villages lived in harmony with nature. But scientific e...

History books traditionally depict the pre-Columbus Americas as a pristine wilderness where small native villages lived in harmony with nature. But scientific evidence tells a very different story: When Columbus stepped ashore in 1492, millions of people were already living there. America wasn't exactly a New World, but a very old one whose inhabitants had built a vast infrastructure of cities, orchards, canals and causeways.
The English brought honeybees to the Americas for honey, but the bees pollinated orchards along the East Coast. Thanks to the feral honeybees, many of the plants the Europeans brought, like apples and peaches, proliferated. Some 12,000 years ago, North American mammoths, ancient horses, and other large mammals vanished. The first horses in America since the Pleistocene era arrived with Columbus in 1493.
Settlers in the Americas told of rivers that had more fish than water. The SouthAmerican potato helped spark a population explosion in Europe. In 1491, the Americas had few domesticated animals, and used the llama as their beast of burden.
In 1491, more people lived in the Americas than in Europe. The first conquistadors were sailors and adventurers. In 1492, the Americas were not a pristine wilderness but a crowded and managed landscape. The now barren Chaco Canyon was once covered with vegetation. Along with crops like wheat, weeds like dandelion were brought to America by Europeans.
It’s believed that the domestication of the turkey began in pre-Columbian Mexico, and did not exist in Europe in 1491. By 1500, European settlers and their plants and animals had altered much of the Americas’ landscape. While beans, potatoes, and maize from the Americas became major crops in continental Europe.

History books traditionally depict the pre-Columbus Americas as a pristine wilderness where small native villages lived in harmony with nature. But scientific evidence tells a very different story: When Columbus stepped ashore in 1492, millions of people were already living there. America wasn't exactly a New World, but a very old one whose inhabitants had built a vast infrastructure of cities, orchards, canals and causeways.
The English brought honeybees to the Americas for honey, but the bees pollinated orchards along the East Coast. Thanks to the feral honeybees, many of the plants the Europeans brought, like apples and peaches, proliferated. Some 12,000 years ago, North American mammoths, ancient horses, and other large mammals vanished. The first horses in America since the Pleistocene era arrived with Columbus in 1493.
Settlers in the Americas told of rivers that had more fish than water. The SouthAmerican potato helped spark a population explosion in Europe. In 1491, the Americas had few domesticated animals, and used the llama as their beast of burden.
In 1491, more people lived in the Americas than in Europe. The first conquistadors were sailors and adventurers. In 1492, the Americas were not a pristine wilderness but a crowded and managed landscape. The now barren Chaco Canyon was once covered with vegetation. Along with crops like wheat, weeds like dandelion were brought to America by Europeans.
It’s believed that the domestication of the turkey began in pre-Columbian Mexico, and did not exist in Europe in 1491. By 1500, European settlers and their plants and animals had altered much of the Americas’ landscape. While beans, potatoes, and maize from the Americas became major crops in continental Europe.

Tyler Hamilton - The true price of winning at all costs

The Discovery Vitality Summit, held from 15-17 August this year, featured a number of keynote speakers, both local and international. One of the main draw cards...

The Discovery Vitality Summit, held from 15-17 August this year, featured a number of keynote speakers, both local and international. One of the main draw cards was Tyler Hamilton, former professional cyclist and teammate of Lance Armstrong. In his talk, Tyler opened up about his journey and the choices he made that led him to become a cycling champion and a user of doping substances. He also fielded questions from the audience about his experience. The footage is an edited version of his talk.

The Discovery Vitality Summit, held from 15-17 August this year, featured a number of keynote speakers, both local and international. One of the main draw cards was Tyler Hamilton, former professional cyclist and teammate of Lance Armstrong. In his talk, Tyler opened up about his journey and the choices he made that led him to become a cycling champion and a user of doping substances. He also fielded questions from the audience about his experience. The footage is an edited version of his talk.

Lick Observatory: Over 100 Years of Discovery

Lick Observatory, the first permanently occupied mountaintop observatory, has been at the forefront of astronomical research for more than 100 years. Enjoy an i...

Lick Observatory, the first permanently occupied mountaintop observatory, has been at the forefront of astronomical research for more than 100 years. Enjoy an insider's tour of Mt. Hamilton's major telescopes and hear Lick's astronomers and astrophysicists talk about the Observatory's remarkable accomplishments - from early discoveries to modern day research. [8/2001] [Science] [Show ID: 5928]

Lick Observatory, the first permanently occupied mountaintop observatory, has been at the forefront of astronomical research for more than 100 years. Enjoy an insider's tour of Mt. Hamilton's major telescopes and hear Lick's astronomers and astrophysicists talk about the Observatory's remarkable accomplishments - from early discoveries to modern day research. [8/2001] [Science] [Show ID: 5928]

While Hillary and Trump distract from serious policy issues that need to be debated in this presidential election, Lyndon LaRouche is the only member of the U.S. Presidency organizing for a Hamiltonian recovery program. See his "The Four New Laws to SaveThe U.S.A. Now! NOT AN OPTION: AN IMMEDIATE NECESSITY"(https://larouchepac.com/four-laws).
Here is an excerpt of our October 14, 2016 webcast featuring MatthewOgden and Ben Deniston discussing the unique genius of Alexander Hamilton and how his approach to growing the economy can be applied to the United States today. Full webcast: https://www.youtube.com/watch?v=jm1nqnvPCEs
----------
Subscribe to LaRouchePAC Live: http://lpac.co/youtube
Subscribe to LaRouchePAC Videos: http://lpac.co/youtube-vid
Subscribe to LaRouchePAC Science: http://lpac.co/youtube-sci
Get active, become an organizer: http://lpac.co/action
Receive daily email updates from LaRouchePAC: http://lpac.co/daily
Donate to LaRouchePAC: http://lpac.co/donate-yt
Keep connected at:
https://larouchepac.com/
https://www.facebook.com/LaRouchePAC
https://soundcloud.com/larouche-pac
https://twitter.com/larouchepac
==========

While Hillary and Trump distract from serious policy issues that need to be debated in this presidential election, Lyndon LaRouche is the only member of the U.S. Presidency organizing for a Hamiltonian recovery program. See his "The Four New Laws to SaveThe U.S.A. Now! NOT AN OPTION: AN IMMEDIATE NECESSITY"(https://larouchepac.com/four-laws).
Here is an excerpt of our October 14, 2016 webcast featuring MatthewOgden and Ben Deniston discussing the unique genius of Alexander Hamilton and how his approach to growing the economy can be applied to the United States today. Full webcast: https://www.youtube.com/watch?v=jm1nqnvPCEs
----------
Subscribe to LaRouchePAC Live: http://lpac.co/youtube
Subscribe to LaRouchePAC Videos: http://lpac.co/youtube-vid
Subscribe to LaRouchePAC Science: http://lpac.co/youtube-sci
Get active, become an organizer: http://lpac.co/action
Receive daily email updates from LaRouchePAC: http://lpac.co/daily
Donate to LaRouchePAC: http://lpac.co/donate-yt
Keep connected at:
https://larouchepac.com/
https://www.facebook.com/LaRouchePAC
https://soundcloud.com/larouche-pac
https://twitter.com/larouchepac
==========

published:17 Oct 2016

views:1214

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Liberty's Kids HD 120 - Alexander Hamilton - An American In Paris | History Cartoons for Children

“There’s a million things I haven’t done, but just you wait, JUST YOU WAIT!"
Subscribe to Liberty's Kids : http://bit.ly/2ivIdi6
Benjamin Franklin enlists the help of young people to record the happenings leading up to and during the Revolution for his newspaper the Pennsylvania Gazette. First, there is a James Hiller, a patriot who tends to act before he thinks and is, at times, too quick a judge in his search for American heroes. Next is Henri, a French orphan who's only quest is for food. Lastly, a young woman named Sarah Phillips joins the team; the daughter of an ex-English general with strong views opposing slavery. Together, they travel the colonies witnessing the sacrifices made for freedom.

“There’s a million things I haven’t done, but just you wait, JUST YOU WAIT!"
Subscribe to Liberty's Kids : http://bit.ly/2ivIdi6
Benjamin Franklin enlists the help of young people to record the happenings leading up to and during the Revolution for his newspaper the Pennsylvania Gazette. First, there is a James Hiller, a patriot who tends to act before he thinks and is, at times, too quick a judge in his search for American heroes. Next is Henri, a French orphan who's only quest is for food. Lastly, a young woman named Sarah Phillips joins the team; the daughter of an ex-English general with strong views opposing slavery. Together, they travel the colonies witnessing the sacrifices made for freedom.

Ross Hamilton has made some incredible discoveries about the Great SerpentMount in Ohio. This ancient Native America earthwork has got a real mystery to reveal. How does the constellation Draco figure into the story, and who, truly, does the serpent represent? Listen as Whitley finds out the answer. Then Linda Howe explores a much more modern mystery: what are those, hard-packed soils scientists see in the recent Mars lander photos?
The Great SerpentMound is a 1,348-foot (411 m)-long, three-foot-high prehistoric effigy mound located on a plateau of the Serpent Mound crater along Ohio Brush Creek in Adams County, Ohio. Maintained within a park by the Ohio Historical Society, it has been designated a National Historic Landmark by the United States Department of Interior. The Serpent Mound of Ohio was first reported from surveys by Ephraim Squire and Edwin Davis in their historic volume Ancient Monuments of the Mississippi Valley, published in 1848 by the newly founded Smithsonian Museum.
Researchers have attributed construction of the mound to three different prehistoric indigenous cultures. Although it was once thought to be Adena in origin, now based on the use of more advanced technology, including carbon dating and evidence from 1996 studies, many scholars now believe that members of the Fort Ancient culture built it about 1070 CE (plus or minus 70 years). There are still anomalies to be studied. Serpent Mound is the largest serpent effigy in the world.
The dating of the design, the original construction, and the identity of the builders of the serpent effigy are three questions still debated in the disciplines of social science, including ethnology, archaeology, and anthropology. In addition, contemporary American Indians have an interest in the site. Several attributions have been entered by academic, philosophic, and Native American concerns regarding all three of these unknown factors of when designed, when built, and by whom.
Over the years, scholars have proposed that the mound was built by members of the Adena culture, the Hopewell culture, or the Fort Ancient culture. In the18th century the missionary John Heckewelder reported that Native Americans of the Lenni Lenape (later Delaware) nation told him the Allegheny people had built the mound, as they lived in the Ohio Valley in an ancient time. Both Lenape and Iroquois legends tell of the Allegheny or Allegewi People, sometimes called Tallegewi. They were said to have lived in the Ohio Valley in a remotely ancient period, believed pre-Adena, i.e., Archaic or pre-Woodland period (before 1200 BCE). Because archaeological evidence suggests that ancient cultures were distinct and separate from more recent historic Native American cultures, academic accounts do not propose the Allegheny Nation built the Serpent Mound.
Recently the dating of the site has been brought into question. While it has long been thought to be an Adena site based on slim evidence, a couple of radiocarbon dates from a small excavation raise the possibility that the mound is no more than a thousand years old. Middle Ohio Valley people of the time were not known for building large earthworks, however; they did display a high regard for snakes as shown by the numerous copper serpentine pieces associated with them.
Radiocarbon dating of charcoal discovered within the mound in the 1990s indicated that people worked on the mound circa 1070 CE.

Ross Hamilton has made some incredible discoveries about the Great SerpentMount in Ohio. This ancient Native America earthwork has got a real mystery to reveal. How does the constellation Draco figure into the story, and who, truly, does the serpent represent? Listen as Whitley finds out the answer. Then Linda Howe explores a much more modern mystery: what are those, hard-packed soils scientists see in the recent Mars lander photos?
The Great SerpentMound is a 1,348-foot (411 m)-long, three-foot-high prehistoric effigy mound located on a plateau of the Serpent Mound crater along Ohio Brush Creek in Adams County, Ohio. Maintained within a park by the Ohio Historical Society, it has been designated a National Historic Landmark by the United States Department of Interior. The Serpent Mound of Ohio was first reported from surveys by Ephraim Squire and Edwin Davis in their historic volume Ancient Monuments of the Mississippi Valley, published in 1848 by the newly founded Smithsonian Museum.
Researchers have attributed construction of the mound to three different prehistoric indigenous cultures. Although it was once thought to be Adena in origin, now based on the use of more advanced technology, including carbon dating and evidence from 1996 studies, many scholars now believe that members of the Fort Ancient culture built it about 1070 CE (plus or minus 70 years). There are still anomalies to be studied. Serpent Mound is the largest serpent effigy in the world.
The dating of the design, the original construction, and the identity of the builders of the serpent effigy are three questions still debated in the disciplines of social science, including ethnology, archaeology, and anthropology. In addition, contemporary American Indians have an interest in the site. Several attributions have been entered by academic, philosophic, and Native American concerns regarding all three of these unknown factors of when designed, when built, and by whom.
Over the years, scholars have proposed that the mound was built by members of the Adena culture, the Hopewell culture, or the Fort Ancient culture. In the18th century the missionary John Heckewelder reported that Native Americans of the Lenni Lenape (later Delaware) nation told him the Allegheny people had built the mound, as they lived in the Ohio Valley in an ancient time. Both Lenape and Iroquois legends tell of the Allegheny or Allegewi People, sometimes called Tallegewi. They were said to have lived in the Ohio Valley in a remotely ancient period, believed pre-Adena, i.e., Archaic or pre-Woodland period (before 1200 BCE). Because archaeological evidence suggests that ancient cultures were distinct and separate from more recent historic Native American cultures, academic accounts do not propose the Allegheny Nation built the Serpent Mound.
Recently the dating of the site has been brought into question. While it has long been thought to be an Adena site based on slim evidence, a couple of radiocarbon dates from a small excavation raise the possibility that the mound is no more than a thousand years old. Middle Ohio Valley people of the time were not known for building large earthworks, however; they did display a high regard for snakes as shown by the numerous copper serpentine pieces associated with them.
Radiocarbon dating of charcoal discovered within the mound in the 1990s indicated that people worked on the mound circa 1070 CE.

FamousMathProbs 13a: The rotation problem and Hamilton's discovery of quaternions I

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

59:47

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

This is the second of three lectures on Hamilton's discovery of quaternions, and here we i...

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

All-New Land Rover Discovery – Surfing with Laird Hamilton

The All-New Land Rover Discovery receives the surfing seal of approval as legendary surfer Laird Hamilton and 8-year old surfing prodigy Jett Prefontaine take the ultimate family SUV on a surf adventure in Malibu. In a display of incredible versatility, smart technology and capability, the All-New Discovery proves why it’s the perfect SUV for any adventure.
Visithttp://bit.ly/2fBmbtf for more All-New Discovery Information.

Alexander Hamilton and Aaron Burr met on the dueling ground one fateful day, but their story started much earlier.
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HISTORY SpecialsSeason 1Episode 1
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1:01:33

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

We show how to practically implement the use of quaternions to describe the algebra of rot...

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

56:14

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

This is the third lecture on the problem of how to extend the algebraic structure of the c...

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

54:56

Hamilton, Boole and their Algebras - Professor Raymond Flood

Professor Flood gives a fabulous overvierw of the lives and work of two mathematicians, Ha...

Hamilton, Boole and their Algebras - Professor Raymond Flood

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

4:47

Forbes Hamilton, Land Rover Discovery

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west...

Forbes Hamilton, Land Rover Discovery

Forbes regularly takes his Land Rover Discovery on 4x4 adventures in the wilds of the west of Scotland, we join him for a fun journey and ask him what else is important in his life. See more at http://crazyway.tv

1:46

20 Discovery Drive, Flagstaff, Hamilton.

For more information please go to https://www.lodge.co.nz/Browse-Properties/Flagstaff/20-D...

FamousMathProbs 13a: The rotation problem and Hamilton's discovery of quaternions I

W. R. Hamilton in 1846 famously carved the basic multiplicative laws of the four dimensional algebra of quaternions onto a bridge in Dublin during a walk with his wife. This represented a great breakthrough on an important problem he had been wrestling with: how to algebraically represent rotations of 3 dimensional space using some kind of analog of complex numbers for rotations of the plane.
This is the first of three lectures on this development, and here we set the stage by introducing complex numbers and explaining some of their natural links with rotations of the plane. There is a lot of information in this lecture, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further. In particular the last slide (page 9) could easily be stared at for an hour or two.
Even old hands at complex analysis may find something novel here to stimulate their thinking, as I insist on a completely logical and rational approach to mathematics--no waffling with angles or ``transcendental notions/functions'' involving ``real numbers''. In fact such a pure algebraic approach is exactly what is needed to set the stage for a good understanding of quaternions.
In particular you will learn that the most fundamental fact about complex numbers is properly stated using the notion of quadrance, that turns are a viable substitute for angles, and that the rational parametrization of a circle is intimately linked to a quadratic map at the level of complex numbers. These ideas will prepare us for appreciating the rotation problem in three dimensions, which we tackle in the next lecture, and then the introduction of quaternions, which we explain in the following one.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

59:47

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

This is the second of three lectures on Hamilton's discovery of quaternions, and here we i...

FamousMathProbs13b: The rotation problem and Hamilton's discovery of quaternions (II)

This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.
Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.
This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

1:01:33

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

We show how to practically implement the use of quaternions to describe the algebra of rot...

FamousMathProbs13d: The rotation problem and Hamilton's discovery of quaternions IV

We show how to practically implement the use of quaternions to describe the algebra of rotations of three dimensional space. The key idea is to use the notion of half-turn instead of angle: this is well suited to connect with the lovely algebraic structure of quaternions.
The theory of half turns is interesting in its own right, and belongs to what we call Vector Trigonometry--an interesting variant of Rational Trigonometry that we intend to describe in detail elsewhere. Here we only need a few formulas for half turns, which really go back to the ancient Greeks and the rational parametrization of the unit circle which we have discussed many times!
By focussing on the formula for quaternion multiplication in terms of scalar and vector parts, we can deduce that any orthonormal set of vectors u,v and w act algebraically just like the familiar unit vectors l,j and k. That allows us to decompose the multiplication of a general quaternion into its action on two perpendicular planes: this is the key to understanding the geometry of quaternion multiplication.
It allows us to easily see the effect of multiplying on the left by q and on the right be the conjugate. After a normalization by the quadrance of q, we get a rotation of the vector part of the space, which is the connection with rotations that we seek.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my ResearchGate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

56:14

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

This is the third lecture on the problem of how to extend the algebraic structure of the c...

FamousMathProbs13c: The rotation problem and Hamilton's discovery of quaternions III

This is the third lecture on the problem of how to extend the algebraic structure of the complex numbers to deal with rotations in space, and Hamilton's discovery of quaternions, and here we roll up the sleaves and get to work laying out a concise but logically clear framework for this remarkable structure.
A main tool that we will use is the algebra of 2x2 matrices, however with (rational) complex number entries. This allows us a simplified way of proving the various laws of arithmetic for quaternions, and brings ideas from linear algebra, like the determinant and the trace of a matrix, into play.
We end with an important visual model of quaternions and the key formula that connects them with rotations of three dimensional space. There is a lot in this lecture, so be prepared to go slowly, take it in bite size pieces if necessary, and try your hand at the problems!
In the next and final lecture on this topic, we will amplify our understanding of the rotation mapping, and show how quaternions can be practically used to realize rotations and their compositions. All without any use of transcendental notions such as angle, cos or sin-- a big step forward in the conceptual understanding of this subject!!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

54:56

Hamilton, Boole and their Algebras - Professor Raymond Flood

Professor Flood gives a fabulous overvierw of the lives and work of two mathematicians, Ha...

Hamilton, Boole and their Algebras - Professor Raymond Flood

ProfessorFlood gives a fabulous overvierw of the lives and work of two mathematicians, Hamilton and Boole: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
William Rowan Hamilton (1805-1865) revolutionized algebra with his discovery of quaternions, a non-commutative algebraic system, as well as his earlier work on complex numbers. George Boole (1815-1864) contributed to probability and differential equations, but his greatest achievement was to create an algebra of logic 'Boolean algebra'. These new algebras were not only important to the development of algebra but remain of current use.
The transcript and downloadable versions of the lecture are available from the Gresham College website: http://www.gresham.ac.uk/lectures-and-events/hamilton-boole-and-their-algebras
Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There are currently over 1,800 lectures free to access or download from the website.
Website: http://www.gresham.ac.uk
Twitter: http://twitter.com/GreshamCollege
Facebook: https://www.facebook.com/greshamcollege
Instagram: http://www.instagram.com/greshamcollege

Fentanyl: The Drug Deadlier than Heroin

VICE presents an immersive and personal feature film about the fentanyl crisis in Canada told from the perspective of a community of drug users.
This video has been edited from its original version due to privacy concerns surrounding one of its subjects.
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1:31:53

America Before Columbus (Full Documentary)

History books traditionally depict the pre-Columbus Americas as a pristine wilderness wher...

America Before Columbus (Full Documentary)

History books traditionally depict the pre-Columbus Americas as a pristine wilderness where small native villages lived in harmony with nature. But scientific evidence tells a very different story: When Columbus stepped ashore in 1492, millions of people were already living there. America wasn't exactly a New World, but a very old one whose inhabitants had built a vast infrastructure of cities, orchards, canals and causeways.
The English brought honeybees to the Americas for honey, but the bees pollinated orchards along the East Coast. Thanks to the feral honeybees, many of the plants the Europeans brought, like apples and peaches, proliferated. Some 12,000 years ago, North American mammoths, ancient horses, and other large mammals vanished. The first horses in America since the Pleistocene era arrived with Columbus in 1493.
Settlers in the Americas told of rivers that had more fish than water. The SouthAmerican potato helped spark a population explosion in Europe. In 1491, the Americas had few domesticated animals, and used the llama as their beast of burden.
In 1491, more people lived in the Americas than in Europe. The first conquistadors were sailors and adventurers. In 1492, the Americas were not a pristine wilderness but a crowded and managed landscape. The now barren Chaco Canyon was once covered with vegetation. Along with crops like wheat, weeds like dandelion were brought to America by Europeans.
It’s believed that the domestication of the turkey began in pre-Columbian Mexico, and did not exist in Europe in 1491. By 1500, European settlers and their plants and animals had altered much of the Americas’ landscape. While beans, potatoes, and maize from the Americas became major crops in continental Europe.

Tyler Hamilton - The true price of winning at all costs

The Discovery Vitality Summit, held from 15-17 August this year, featured a number of keynote speakers, both local and international. One of the main draw cards was Tyler Hamilton, former professional cyclist and teammate of Lance Armstrong. In his talk, Tyler opened up about his journey and the choices he made that led him to become a cycling champion and a user of doping substances. He also fielded questions from the audience about his experience. The footage is an edited version of his talk.

28:52

"Backroads of Montana: Episode 7 - Havre to Hamilton" (1995)

Episode #7 of "Backroads of Montana" hosted by Montana TV & Radio personality, William Mar...

Linda Hamilton ile Investigation Discovery belgese...

Incredible Discoveries about the Great Serpent Mou...

In August 2016, a research plane was able to observe something strange in the atmosphere above Alaska's Aleutian Islands, lingering aerosol particle that was enriched with the same kind of uranium used in nuclear fuel and bombs, according to Gizmodo. The observation was the first time that scientists detected a particle free-floating in the atmosphere in over 20 years of plane-based observations ... ... -WN.com, Maureen Foody....

ADDIS ABABA, Ethiopia (AP) -- Ethiopia's defense minister on Saturday ruled out a military takeover a day after the East African nation declared a new state of emergency amid the worst anti-government protests in a quarter-century. The United States said it "strongly disagrees" with the new declaration that effectively bans protests, with a U.S ... He also ruled out a transitional government ... Learn more about our and . ....

One day in August 1995 a man called Foutanga Babani Sissoko walked into the head office of the Dubai Islamic Bank and asked for a loan to buy a car. The manager agreed, and Sissoko invited him home for dinner. It was the prelude, writes the BBC's Brigitte Scheffer, to one of the most audacious confidence tricks of all time. Over dinner, Sissoko made a startling claim ... With these powers, he could take a sum of money and double it ... ....

MEXICOCITY. A strong earthquake shook southern and central Mexico Friday, causing panic less than six months after two devastating quakes that killed hundreds of people. No buildings collapsed, according to early reports. But two towns near the epicenter, in the southern state of Oaxaca, reported damage and state authorities said they had opened emergency shelters ... It was also felt in the states of Guerrero, Puebla and Michoacan ... AFP ... ....

Mexico City – A military helicopter carrying officials assessing damage from a powerful earthquake crashed Friday in southern Mexico, killing 13 people and injuring 15, all of them on the ground. The Oaxaca state prosecutor’s office said in a statement that five women, four men and three children were killed at the crash site and another person died later at the hospital ...Alejandro Murat, neither of whom had serious injuries ... The U.S ... ....

COMPETITIVE CHEERHamilton, WestOttawa advanceBoth the Hamilton and West Ottawa competitive cheerleading teams both finished fourth in their respective district tournaments on Saturday to advance to the regional tournament on Saturday, Feb. 24.West Ottawa finished fourth in District 1-1 at East Kentwood with 758.26 points ...The Panthers will compete in Brighton while Hamilton heads to Rockford.Hamilton finished ... ....

HAMILTON — Two people were able to safely escape a raging fire that gutted a home on North10th Street Friday night, according to Hamilton fire officials. Fire crews responded to a fire in progress at 5.47 p.m. at 327 N. 10th St ... ....

We’ve known for a year that the major Broadway smash “Hamilton” would be presented at the Durham Performing Arts Center... The touring version of “Hamilton” will have a nearly month-long run from Nov ... Now, how does one get a ticket to “Hamilton”? The one guaranteed way is if you’re a subscriber of DPAC’s 2018-19 Broadway season, on sale now to current 2017-18 season ticket-holders ... ▪ “Hamilton”....

AgendaWaikato gathered $20,000 in funding for entertainment at the Victoria on the River park on Wednesday after HamiltonCity Council voted down the idea the week before ... * No money for entertainment in Hamilton's new park, Victoria on the River. * Victoria on the River opens in Hamilton. * Budget chop wins support for Hamilton urban park plan....

After consolations the Pikes Peak region has three teams in the top 10 in Class 4A. Mesa Ridge sits in fourth place with 82.5 points and two wrestlers in the finals. MichaelTrue will battle for gold in 285, while ElijahValdez will look for a win in 152. DiscoveryCanyon sits in sixth with 70 points and one wrestler in the finals. Patrick Allis advanced to the finals in 120... Both teams have one wrestler still alive ... ....